Thermal Conduction, Thermal Conductivity – Problems and Solutions – Laman 5

HOTS

05. The figure shows a perfectly insulated copper rod with a thermal conductivity of 380 W m^-1 K^-1. The diameter of the rod is 5 cm. The temperature at P and Q are maintained at 60⁰C and 45⁰C respectively. The rod has reached steady state.

Find
a. the temperature gradient
b. the rate of heat flow
c. the temperature at a point 10 cm away from Q
d. the amount of heat flowing through P and Q in 12 minutes.

Solution

a.
$\frac { d\theta }{ dx } =\quad \frac { (45\quad -\quad 60) }{ 25 } \\ \\ \frac { d\theta }{ dx } =\quad -{ 0.6 }^{ 0 }C\quad c{ m }^{ -1 }\\ \\ \frac { d\theta }{ dx } =\quad -{ 60 }^{ 0 }C\quad { m }^{ -1 }\\ \\ \\ \\$

b.
$\frac { dQ }{ dx } =\quad -kA\frac { d\theta }{ dx } \\ \\ \frac { dQ }{ dx } =\quad -380\pi (\frac { { 0.05 }^{ 2 } }{ 4 } )(-60)\quad J{ s }^{ -1 }\\ \\ \frac { dQ }{ dx } =\quad 44.77J{ s }^{ -1 }$

c. Let R be the point 10 cm away from Q. Temperature at
$R\quad =\quad 60\quad +\quad [-0.6\quad x\quad (25-10)]\\ \\ R\quad =\quad 60\quad +\quad [-0.6\quad x\quad (15)]\\ \\ R\quad =\quad 60\quad +\quad (-9)\\ \\ R\quad =\quad { 51 }^{ 0 }C\\ \\$

d. In steady state,
$\frac { dQ }{ dt } =\frac { Q }{ t } =44.77\\ \\ Q\quad =\quad 44.77t\quad \\ \\ Q\quad =\quad 44.77\quad (12x60)\quad J\\ \\ Q\quad =\quad 32234.4J$